4 SPECTRUM OF A COMPACT OPERATOR 327

(d) Conversely, let A ^ 0 be in sp(w). Show that for each n there is (at least) a number
pn e sp(tfn) such that A = lim pn. (Otherwise, one can assume that there is an open

n->oo

disk D of center A and radius r, such that D n sp(w) = {A} and D n sp(«n) = 0
(extract from («„) a suitable subsequence). Let then y be the road t -> A -f re** defined
in [0, 2-rr]; consider the integral

,-£-lB>-l<£-A)*d£«-0 for *>0

and use (b) to obtain a contradiction.)

(e) Let A =£ 0 be in sp(«), and let D be an open disk of center A and radius r such that
D n sp(«)= {A}; there exists n0 such that, for n ^ n0 the intersection of sp(z/n) and
of the circle |f — A| = r is empty (use (c)). Let /xl9..., fjir be the points of D n sp(z/n),

r

and write kn = £ M^; */„)• Show that there exists «i such that, for TI > nl9 kn ^ &(A; w).

(Use the same method as in (d), multiplying (un — I • IE)""* by a suitable polynomial in
£ of degree kn.) Give an example in which kn > k(X; it) for every n.

r

(f) With the notations of (e), let /? =/u,«, At = Z^/,un; show that lim/>„=/>

/=! n-t-oo

in the Banach space <&(E) (use (b), and Problem 1). Deduce from that result that there

exists /22 such that, for n ^ n2, Nn — N(/AI ; WM) +-----h N(/A, ; «„) is a supplement to

F(A; w) in E. (Suppose n is such that \\p — pn \\ ^ 1/2; if there was a point xn e F(A) r\ Nn
such that ||jcn||=l, then the relations p(xn) — 0, pn(xn) = *„ would contradict the
preceding inequality. Prove similarly that the intersection of N(A; u) and of the sub-
space F(/ZI; un) n >- r\ F(ju-r; ww) is reduced to 0.)

3. Let u be a compact operator in an infinite dimensional complex Banach space E,
and let P(£) be a polynomial without constant term; put U = P(M). Show that the
spectrum sp(u) is identical to the set of numbers P(A), where A esp(w); furthermore,
for every ju, e sp(u), N(/x; v) is the (direct) sum of the subspaces N(Ak; M) such that
P(Afc) = /x, and F(/x; v) the intersection of the corresponding subspaces F(Afc; u). (Let
V be any closed subspace of E, such that w(V) <= V, and let wv be the restriction of
u to V. Show that there is a constant M independent of V and «, such that ||(P(wv))n II ^
Mw||«v||. Apply that remark and Problem 1 of Section 11.1, taking for V a suitable
intersection of a finite number of subspaces of the form F(A; w).)

4. Let E be a separable Hilbert space, (en)n^o a Hilbert basis of E. Show that the operator
u defined by u(en) = en+i/(n ~f- 1) for n ^ 0 is compact and that sp(w) is reduced to 0
(more precisely, u has no eigenvalue).

5. Let u be a continuous operator in a complex Banach space E. A Rfesz point for u is
a point A in the spectrum sp(w) such that: (1) A is isolated in sp(«); (2) E is the direct
sum of a closed subspace F(A) and of a finite dimensional subspace N(A) such that
w(F(A)) c F(A), w(N(A)) <= N(A), the restriction of u - A • 1E to F(A) is a linear homeo-
morphism and the restriction of u — A • 1E to N(A) is nilpotent.

(a) If A and /x are two distinct Riesz points in sp(w), show that N(ju) <=; F(A), and
F(A) is the direct sum of N(/z) and F(A) n F(ft).

(b) A Riesz operator u is denned as a continuous operator such that all points ^ 0
in the spectrum sp(u) are Riesz points. For any e > 0, the set of points A e sp(u) such
that |A| ^ B is then a finite set {p,lt..., /x,.}; let pt be the projection of E onto N(/xf)
in the decomposition of E into the direct sum N(/*j) + F(/x./) (1 =^ i* < r), and let

v^u— J\u°pt. Show that sp(v) is contained in the disk |£| <e, hence (Section 11.1,

iti

Problem 1) that lim ||i>"||1/n< e.efined in I x R. Let S be an open ball of center x0 in E, and let